# Definition:Jordan Arc

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## Definition

Let $\left({x_1, y_1}\right), \left({x_2, y_2}\right) \in \R^2$.

Let $f: \left[{0 \,.\,.\, 1}\right] \to \R^2$ be a path from $\left({x_1, y_1}\right)$ to $\left({x_2, y_2}\right)$.

Then $f$ is a **Jordan arc** if and only if $f$ is an injection, except that we allow the possibility $\left({x_1, y_1}\right) = \left({x_2, y_2}\right)$.

## Also defined as

In many sources, a **Jordan arc** $f$ is defined as a path that is an injection, so the initial point of $f$ is different from the final point of $f$.

That is, $f$ is a homeomorphism of the closed unit interval $\left[{0 \,.\,.\, 1}\right]$.

## Also see

## Source of Name

This entry was named for Marie Ennemond Camille Jordan.